Why don't these limits exist? Why don't these limits exist?
$$\lim_{x\to 0}\frac1{x^2}$$
Aren't we here just approaching $x$? And, thus will not get $0$ and avoid the division on "0"?
And, what about this limit?
$$\lim_{x\to 0}\frac{|x|}{x}$$
 A: In these instances, the phrasing 'does not exist' really has two related meanings, one for each of your examples.


*

*for the first example, whether you go from the left or the right, you approach a very large number (what is usually called infinity' or '$\infty$'). And this 'number' (it's a bit questionable to call it a 'number') is not manipulable by our usual means (it follows some rules, bu not all. So, to avoid allowing use of it afterwards, we say 'it does not exist'

*for the second example, the limit value is different when going from opposites directions. Since there's no good way to choose between them, and we really don't think of passing along pairs of numbers, we just say 'the limit does not exist'. 
So in some sense these answers, $\infty$ and the pair {$\infty$ and $-\infty$), exist, but they are not really usable by our usual algebraic manipulations, so we preclude worrying about them by saying 'they don't exist'.
I have a tendency to say that the first example is $\infty$ (because there are instances where you can manipulate that as a value) and the second is 'undefined' (because it is a bit harder to manipulate a function that sometimes returns two values, when really you want only one). 
A: The problem with the first one is not division by zero. The problem with the first one is that as $x$ approaches $0$, the value $\frac{1}{x^2}$ gets larger in size; it can get larger than anything you care to specify ahead of time, by simply taking $x$ small enough. So although we never have to actually divide by $0$, the problem is that the values are not approaching anything: they are "going to $\infty$" (meaning, growing without limit).
A: It's not clear to me what you intended the first limit expression to be.  In the case of the second one, consider the graph of $y=\frac{|x|}{x}$:

As $x\to 0^+$ (that is, as $x$ goes to $0$ from the positive/right side), $\frac{|x|}{x}\to 1$, but as $x\to 0^-$ (that is, as $x$ goes to $0$ from the negative/left side), $\frac{|x|}{x}\to -1$.  Since these two "one-sided" limits do not agree, we say that $$\lim_{x\to 0}\frac{|x|}{x}$$ does not exists.
A: $f(x)\to\infty$ as $x\to0$ if we can find some $k\gt0$ such for any $A\gt0$, $f(x)
\gt A$ when $|x|\lt k$ from this the answer for the fist one is $f(x)\to\infty$ (the limit exists in this particular definition)..see Differential Calculus by Ferrar
